Applications

Contents

Definition

A subfunctor of a functorG:C→DG:C\to D between categoriesCC and DD is a pair (F,i)(F,i) where F:C→DF:C\to D is a functor and i:F→Gi:F\to G is a natural transformation such that its components iM:F(M)→G(M)i_M:F(M)\to G(M) are monic.

Properties

In a concrete category with images one can choose a representative of a subfunctor where the components of ii are genuine inclusions of the underlying sets; then a subfunctor is just a natural transformation whose components are inclusions. The naturality in terms of concrete inclusions just says that for all f:c→df:c\to d, F(f)=G(f)|F(c)F(f)=G(f)|_{F(c)}. If the set-theoretic circumstances allow consideration of a category of functors, then a subfunctor is a subobject in such a category.

A subfunctor (F,i)(F,i) of the identityidC:C→Cid_C:C\to C in a category with images is an often used case: it amounts to a natural assignment c↦F(c)↪icc\mapsto F(c)\stackrel{i}\hookrightarrow c of a subobject to each object cc in CC. For concrete categories with images then F(f)=f|F(c)F(f)=f|_{F(c)}.